×

Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. (English) Zbl 0842.35019

Most of the results in the paper are aimed at the characterization of the maximal viscosity subsolutions \(u \in USC (\overline \Omega)\) of the Hamilton-Jacobi equation \[ H \bigl( Du(x) \bigr)= f(x),\;x \in \Omega, \quad u(x) \leq b(x),\;x \in \partial \Omega \tag{1} \] as a viscosity supersolution of the related equation \[ {H \bigl( Du (x) \bigr) \over f(x)} = 1, \quad x \in \Omega_0 : = \Omega \backslash \{0\}, \;u(x) \leq b(x), \;x \in \partial \Omega \tag{2} \] under suitable assumptions on the data \(\Omega\), \(H\), \(f\) and \(b\).
Obtaining first a comparison principle, the authors prove that if \(b \in USC (\partial \Omega)\) and if the set \(S(b) \) of the viscosity subsolutions of (1) is not empty then the function \(u(x) : = \sup \{w(x); w(\cdot) \in S(b)\}\) is a locally-Lipschitz viscosity supersolution of (2).
Similar results are proved for some non-autonomous equations and are applied to the value functions of the exit-time problem in optimal control.

MSC:

35F30 Boundary value problems for nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen Y.G., J. Diff. Geom. 33 pp 749– (1991)
[2] Crandall M.G., Bull. Amer. Math. Soc. 27 pp 1– (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[3] Crandall M.G., Trans. Amer. Math. Soc. 277 pp 1– (1983) · doi:10.1090/S0002-9947-1983-0690039-8
[4] Evans L.C., J. Diff. Geom. 33 pp 635– (1991)
[5] Goto S., Diff. Int. Equations 7 pp 323– (1994)
[6] Ishii H., Proc. Amer. Math. Soc. 100 pp 247– (1987) · doi:10.1090/S0002-9939-1987-0884461-3
[7] H. Ishii, Degenerate parabolic PDEs with discontinuities and evolutions of surfaces, in preparation. · Zbl 0841.35057
[8] H. Ishii and P.E. Souganidis, Generalized motion of noncompact hyper-surfaces with velocity having arbitrary growth on the curvature tensor, to apear in Tôhoku Math. J., 47 (1995).
[9] Lions P.L., Research Notes in Math. 69 (1982)
[10] M. Ohnuma and K. Sato, in preparation.
[11] Ohnuma M., Diff. Int. Equations 6 pp 1265– (1993)
[12] Siconolfi A., Commun. Partial Differential Equations 20 pp 277– (1995) · Zbl 0814.35012 · doi:10.1080/03605309508821094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.